Colleges in Rajasthan

No, you won't be eligible for taking admission in any of the B-Schools without appearing in any of the competitive exams like mat, CMAT, CAT, XAT, CET etc. but yes if you are thinking to pursue distance MBA than no competitive exams are needed.

It depends on the syllabus of different Universities. Although 3 years degree course in 6 semesters.

It is better to secure your seat in SRM.

No, there is no such provision at BITS.

NMIMS is the best college.

Admission is through counselling based on the score obtained in Common Entrance Test (CUCET). Eligible candidates need to appear for Central Universities Common Entrance Test (CUCET). Please refer to the link provided below for eligibility and other details to pursue this course from Central University of Rajasthan.
https://www.shiksha.com/teaching-education/course/integrated-m-sc-b-ed-in-physics-central-univeristy-of-rajasthan-ajmer-268877

Yes, you are eligible. First of all you have to clear BETET for Bihar and JTET for Jharkhand to apply for the vacancies in certain government schools.